The given integral is:

$I = \int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} \,dx$

Step 1: Substitution

Let $u = \sqrt{x}$, so that:

$du = \frac{1}{2\sqrt{x}}dx$

$dx = 2u \, du$

Since $\sqrt{x} = u$, the limits change as follows:

• When $x = 1$, then $u = \sqrt{1} = 1$.
• When $x = 4$, then $u = \sqrt{4} = 2$.

Step 2: Rewrite the Integral

Substituting $x$ in terms of $u$:

$I = \int_{1}^{2} \frac{e^u}{u} \cdot 2u \, du$

$I = \int_{1}^{2} 2e^u \, du$

Step 3: Evaluate the Integral

$I = 2 \int_{1}^{2} e^u \, du$

$I = 2 \left[ e^u \right]_{1}^{2}$

$I = 2 (e^2 - e^1)$

$I = 2(e^2 - e)$

Final Answer:

$\mathbf{2(e^2 - e)}$

Would you like a further breakdown or verification?

Related Questions:

1. How do you apply substitution to solve definite integrals?
2. What is the importance of changing limits in definite integrals?
3. How does the exponential function simplify integration?
4. What are common substitutions for integrals involving square roots?
5. Can this integral be evaluated using numerical methods?

Tip:

Always remember to change the limits when performing substitution in definite integrals.